Order parameters transitional

From a discussion of the limiting cases q/1 l.-> o, i l./q 0 the chain length dependence of the critical properties (order parameter, transition temperature) for a constant value of q can be discussed, for example initial increase and then flattening at high degree of polymerization of the transition temperature T was shown to occur. This has indeed been found in many liquid crystalline polymers ". ... 

For the kind of transition above which the order parameter is zero and below which other values are stable, the coefficient 2 iiiust change sign at the transition point and must remain positive. As we have seen, the dependence of s on temperature is detemiined by requiring the free energy to be a miniimuii (i.e. by setting its derivative with respect to s equal to zero). Thus... 

A feature of a critical point, line, or surface is that it is located where divergences of various properties, in particular correlation lengths, occur. Moreover it is reasonable to assume that at such a point there is always an order parameter that is zero on one side of the transition and tliat becomes nonzero on the other side. Nothing of this sort occurs at a first-order transition, even the gradual liquid-gas transition shown in figure A2.5.3 and figure A2.5.4. 

If the scalar order parameter of the Ising model is replaced by a two-component vector n = 2), the XY model results. All important example that satisfies this model is the 3-transition in helium, from superfiuid helium-II... 

Here we shall consider two simple cases one in which the order parameter is a non-conserved scalar variable and another in which it is a conserved scalar variable. The latter is exemplified by the binary mixture phase separation, and is treated here at much greater length. The fonner occurs in a variety of examples, including some order-disorder transitions and antrferromagnets. The example of the para-ferro transition is one in which the magnetization is a conserved quantity in the absence of an external magnetic field, but becomes non-conserved in its presence. 

For a one-component fluid, the vapour-liquid transition is characterized by density fluctuations here the order parameter, mass density p, is also conserved. The equilibrium structure factor S(k) of a one component fluid is... 

Flere we discuss only briefly the simulation of continuous transitions (see [132. 135] and references therein). Suppose that tire transition is characterized by a non-vanishing order parameter X and a corresponding divergent correlation length We shall be interested in the block average value where the L... 

The Maier-Saupe tlieory was developed to account for ordering in tlie smectic A phase by McMillan [71]. He allowed for tlie coupling of orientational order to tlie translational order, by introducing a translational order parameter which depends on an ensemble average of tlie first haniionic of tlie density modulation noniial to tlie layers as well as / i. This model can account for botli first- and second-order nematic-smectic A phase transitions, as observed experimentally. 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

Using this order parameter, the free energy in the nematic phase close to a transition to the smectic phase can be shown to be given by [20, 88, 89, 91]... 

This transition is usually second order [18,19 and 20]. The SmC phase differs from the SmA phase by a tilt of the director with respect to the layers. Thus, an appropriate order parameter contains the polar (0) and azimuthal ((]i) angles of the director ... 

The transition from smectic A to smectic B phase is characterized by tire development of a sixfold modulation of density witliin tire smectic layers ( hexatic ordering), which can be seen from x-ray diffraction experiments where a sixfold symmetry of diffuse scattering appears. This sixfold symmetry reflects tire bond orientational order. An appropriate order parameter to describe tlie SmA-SmB phase transition is tlien [18,19 and 20]... 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

Thermal stahility. Yor applications of LB films, temperature stability is an important parameter. Different teclmiques have been employed to study tliis property for mono- and multilayers of arachidate LB films. In general, an increase in temperature is connected witli a confonnational disorder in tire films and above 390 K tire order present in tire films seems to vanish completely [45, 46 and 45] However, a comprehensive picture for order-disorder transitions in mono- and multilayer systems cannot be given. Nevertlieless, some general properties are found in all systems [47]. Gauche confonnations mostly reside at tire ends of tire chains at room temperature, but are also present inside tire... 

There are transition temperatures in some Hquid crystals where the positional order disappears but the orientational order remains (with increasing temperature). The positional order parameter becomes zero at this temperature, but unlike i, this can either be a discontinuous drop to zero at this temperature or a continuous decrease of the order parameter which reaches zero at this temperature. 

With increasing values of P the molar volume is in progressively better agreement with the experimental values. Upon heating a phase transition takes place from the a phase to an orientationally disordered fee phase at the transition temperature where we find a jump in the molar volume (Fig. 6), the molecular energy, and in the order parameter. The transition temperature of our previous classical Monte Carlo study [290,291] is T = 42.5( 0.3) K, with increasing P, T is shifted to smaller values, and in the quantum limit we obtain = 38( 0.5) K, which represents a reduction of about 11% with respect to the classical value. 

The central quantity is the order parameter as a function of temperature (see Fig. 13). The phase transition temperature Tq of the classical system can be located around 38 K. At high temperatures, the quantum curve of the order parameter merges with the classical curve, whereas it starts to deviate below Tq. Qualitatively, quantum fluctuations lower the ordering and thus the quantum order parameter is always smaller than its classical counterpart. The inclusion of quantum effects results in a nearly 10% lowering of Tq (see Fig. 13). 

Furthermore, one can infer quantitatively from the data in Fig. 13 that the quantum system cannot reach the maximum herringbone ordering even at extremely low temperatures the quantum hbrations depress the saturation value by 10%. In Fig. 13, the order parameter and total energy as obtained from the full quantum simulation are compared with standard approximate theories valid for low and high temperatures. One can clearly see how the quasi classical Feynman-Hibbs curve matches the exact quantum data above 30 K. However, just below the phase transition, this second-order approximation in the quantum fluctuations fails and yields uncontrolled estimates just below the point of failure it gives classical values for the order parameter and the herringbone ordering even vanishes below... 

Here we review the properties of the model in the mean field theory [328] of the system with the quantum APR Hamiltonian (41). This consists of considering a single quantum rotator in the mean field of its six nearest neighbors and finding a self-consistent condition for the order parameter. Solving the latter condition, the phase boundary and also the order of the transition can be obtained. The mean-field approximation is similar in spirit to that used in Refs. 340,341 for the case of 3D rotators. 

Moreover, the order parameter, which in the case of the gas-liquid transition is defined as the difference between the densities of both coexisting phases, A6 = 62 — 61, approaches zero when the temperature goes to (from below, since above T. the above order parameter is always equal to zero) as... 

The seeond-order IPX is fairly well understood. In faet, it has been eon-jeetured that this kind of transition, eharaeterized by an sealar order parameter, given in this example by the eoneentration of the minority speeies... 

M. Allain, P. Oswald, J. M. di Meglio. Structural defects and phase transition in a lyotropic system optical birefringence and order parameter measurements. Mol Cry St Liq Cryst 7625 161-169, 1988. 

Before trying to solve the master equation for growth processes by direct stochastic simulation it is usually advisable to first try some analytical approximation. The mean-field approximation often gives very good results for questions of first-order phase transitions, and at least it provides a qualitative understanding for the interplay of the various model parameters. 

It is not an easy task to develop computer codes which correctly treat the advancement of a folding interface as a boundary condition to a diffusion or flow field. In addition, the interface between a solid and a liquid, for example, is usually is not absolutely sharp on an atomic scale, but varies over a few lattice constants [32,33]. In these cases, it is sometimes convenient to treat the interface as having a finite non-zero thickness. An order parameter is then introduced, which for example varies from the value zero on one side of the interface to the value one on the other, representing a smooth transition from liquid to solid across the interface. This is called the phase-field... 

Those Warren-Cowley parameters have been determined in situ above the order-disorder transition temperature by diffuse neutron scattering. From these experimentally determined static correlations, the first nine effective pair interactions have been deduced using inverse Monte Carlo simulations. 

We display in Fig. 2 some 2-d order-parameter and spin maps at different temperatures. These pictures already gives us a qualitative informations on the thermodynamical behavior of the APB. The APB appears to be fiat for T=1000 K and 1400 K, while it seems to be rough at T=1550 K. It becomes even more rough at higher temperature T=1675 K, as shown in Fig. 3. This visual analysis shows us, without any ambiguity, that the APB does indeed undergo a roughening transition as the temperature increases. 

Specializing to symmetrical mixtures (Na = Nb = N) for simplicity, it is easy to also find the order parameter i ocx of the unmixing transition... 

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